Step | Hyp | Ref
| Expression |
1 | | elxp 5657 |
. . 3
⊢ (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴))) |
2 | | df-rex 3071 |
. . . . . 6
⊢
(∃𝑦 ∈
𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩))) |
3 | | an13 646 |
. . . . . . 7
⊢ ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩))) |
4 | 3 | exbii 1851 |
. . . . . 6
⊢
(∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩))) |
5 | 2, 4 | bitr4i 278 |
. . . . 5
⊢
(∃𝑦 ∈
𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴))) |
6 | | elsni 4604 |
. . . . . . . . 9
⊢ (𝑥 ∈ {𝑋} → 𝑥 = 𝑋) |
7 | 6 | opeq1d 4837 |
. . . . . . . 8
⊢ (𝑥 ∈ {𝑋} → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩) |
8 | 7 | eqeq2d 2744 |
. . . . . . 7
⊢ (𝑥 ∈ {𝑋} → (𝑍 = ⟨𝑥, 𝑦⟩ ↔ 𝑍 = ⟨𝑋, 𝑦⟩)) |
9 | 8 | biimpa 478 |
. . . . . 6
⊢ ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩) |
10 | 9 | reximi 3084 |
. . . . 5
⊢
(∃𝑦 ∈
𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) |
11 | 5, 10 | sylbir 234 |
. . . 4
⊢
(∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) |
12 | 11 | exlimiv 1934 |
. . 3
⊢
(∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) |
13 | 1, 12 | sylbi 216 |
. 2
⊢ (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) |
14 | | snidg 4621 |
. . . . 5
⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋}) |
15 | | opelxpi 5671 |
. . . . 5
⊢ ((𝑋 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴)) |
16 | 14, 15 | sylan 581 |
. . . 4
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴)) |
17 | | eleq1 2822 |
. . . 4
⊢ (𝑍 = ⟨𝑋, 𝑦⟩ → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))) |
18 | 16, 17 | syl5ibrcom 247 |
. . 3
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴))) |
19 | 18 | rexlimdva 3149 |
. 2
⊢ (𝑋 ∈ 𝑉 → (∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴))) |
20 | 13, 19 | impbid2 225 |
1
⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)) |